ABC analysis is a useful method to classify inventory items into different categories that can be specifically managed and controlled. Traditional ABC analysis classifies inventory items into three categories according to the annual dollar usage of an inventory item. Multi criteria inventory classification models have been proposed by researchers in order to take into account other important criteria. From these models we considered in this paper the R- model and the Ng-model in order to improve them. Illustrative examples are presented to compare the improved models and the initial ones.

1. IOSR Journal of Business and Management (IOSR-JBM)
e-ISSN: 2278-487X, p-ISSN: 2319-7668. Volume 11, Issue 6 (Jul. - Aug. 2013), PP 21-27
www.iosrjournals.org
www.iosrjournals.org 21 | Page
An improvement of two multi-criteria inventory classification
models
Makram Ben Jeddou
D´epartement Sciences Economiques et Gestion, Institut Sup´erieur des Etudes Technologiques de Rad`es,
BP 172-2098 Rad`es M´edina-Tunisie.
Abstract: ABC analysis is a useful method to classify inventory items into different categories that
can be specifically managed and controlled. Traditional ABC analysis classifies inventory items into three
categories according to the annual dollar usage of an inventory item. Multi criteria inventory
classification models have been proposed by researchers in order to take into account other important
criteria. From these models we considered in this paper the R- model and the Ng-model in order to
improve them. Illustrative examples are presented to compare the improved models and the initial ones.
Keywords: ABC classification, Multi criteria inventory classification models, R-model, Ng-model.
I. Literature review
Companies often manage a large number of items. It is difficult to give all these items, the same
level of monitoring and control. For this reason, managers should set priority levels for each category
of items. Once this hierarchy established, we can assign specific business rules to each family of items.
In this field, the ABC classification is mostly used by managers for inventory control. The classical
ABC classification is based on the Pareto principle and according to the criterion of the annual use
value. In fact, Class A items are relatively few in number (between 10% and 20%) but they represent
between 70% and 80% of the total annual use value. The Class A items represents a great annual use
value. For this reason, control of items of this class must be rigorous and tight. Class B contains
between 30% and 40% of items that represent 15% to 20% of the total annual use value. The inventory
control of class B items can be less tight than that of the first class A. Finally class C items can reach 50%
of the items but represent only 5% to 10% in terms of value. Managers can reduce their effort in
controlling these items because they are numerous but too small in terms of total annual use value.
When handling inventory classification, managers need to consider at the same time many
important classification factors. Several Methodologies have been proposed in the literature for the multi-
criteria inventory classification problem (MCIC), but we will focus on the weighted optimization
models. These models used a weighted additive function to aggregate the performance of an inventory
item in terms of different criteria to a single score, called the inventory score of an item.
Ramanathan (2006) proposed a linear model which gives each item an opportunity to choose a set of
weights favorable to the item itself when maximizing its score. To improve this model proposed by
Ramanathan (2006), Zhou and Fan (2007) suggested a new classification based on a composite index.
This score or index is a combination of a good index (maximizing score) and a bad index (minimizing
score). The items classification will be then based on the calculated composite index. Ng (2007) proposed
a linear programming model for the MCIC using a scalar measuring for all the criteria and with the
main contribution of giving an importance order to the criteria. Hadi-Vencheh (2010) presented a
nonlinear programming model as an extension version of the Ng-model. This model changed in the linear
programming the constraint of the criterion weights by considering squared coefficients. Finally, Chen
(2011) developed an improved approach to MCIC by which all items are peer-estimated. This new approach
first determines two performance scores for each inventory item, then we determine a weight for each
score in order to calculate a final aggregated index that will be used for the classification.
In this paper we will suggest firstly, a modified variant of R-model proposed by Ramanathan
(2006) for MCIC. In this version, called R*-model, we consider the normalization step of the data to
measure and analyze the impact of this pre-processing on the initial model. In the second part of this
work, we will propose a new weighted linear optimization model which is an improvement of the Ng-
model developed by Ng (2007). This new model will be referred to as M-Ng model.

2. An improvement of two multi-criteria inventory classification models
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II. Impact of normalization step on the R-model
In this section, we start by recalling the R-model.
1,2......Jj0w
1,2.....Nn1
ij
1
1







J
j
njij
J
j
ijiji
yw
tosubject
ywMaxS
The main objective of the normalization step is to eliminate the bias due to the unit effect. This
transformation is considered as a pre-processing step in the weighted optimization multicriteria inventory
models. This step plays a crucial role in standardizing scale measures of the yij  i = 1, ...., n;  j = 1, ......J
where yij is the performance of the item i on the criterion j. More formally, this step convert all the yij in a 0
− 1 scale. Theyij  i = 1, ...., n;  j = 1, ......J , will be transformed on y*ij  i = 1, ...., n;  j = 1, ......J
by using the following expression :
 
   ijNiijNi
ijNiij
ij
yy
yy
y
,...1,...1
,...1*
minmax
min





The proposed R*-model will then consider the normalized mesures y*ij instead of the initials yij as follows:
1,2......Jj0w
1,2.....Nn1
ij
1
*
*
1







J
j
njij
ij
J
j
iji
yw
tosubject
ywMaxS
III. Modified Ng-Model
In this section firstly we recall the Ng-model proposed by Ng (2007). Then we detailed the
modifications made on this model in order to improve it.
III.1. The Ng-Model
To improve the quality of the inventory process, decision makers are needed to classify items
using sophisticated and adequate methods. Generally these methods are based on advanced optimization
backgrounds. In order to help the inventory managers in his inventory control and make it more easier,
Ng(2007) proposes a multicriteria inventory classification model with a minimal mathematical background and
without the use of an optimization solver. In fact Ng-model provides the scores of inventory items
without a linear optimizer.
Consider an inventory with n items must be classified based on J criteria.
yij : the measurement of the ith
item under the jth
criterion.
wij : the non-negative weight represents the contribution of performance of the ith
item for the jth
criterion.
xij : the sum of the first jth
yij of the item i.
Si : the score of the item i.
zi : the dual variable.

3. An improvement of two multi-criteria inventory classification models
www.iosrjournals.org 23 | Page
J...,1,2.......j0w
1-J1,2......,j0w
1
ij
)1(ij
1
1









ji
J
j
ij
J
j
ijiji
w
w
tosubject
ywMaxS
After two transformations of the Ng-Model, we obtain:
III.2. Modified Ng-model (M-Ng model)
The modified Ng-model is a two-phase model. The first phase is very similar to the original model but
we remove the dominance order between the criteria. More precisely, we eliminate the following constraint
in the linear model : wij − wi(j−1) ≥ 0 in order to avoid the subjectiveness or in case of no consensus to set
dominance order relations.
The second phase consists, firstly in computing the average weight of each criterion using weights
generated by the original Ng-model. Secondly we use these obtained average weights to compute final score
for each item. Finally, we classify items according to ABC principle and satisfying the constraint distribution.
This second phase try to overcome the weakness of Ng-model in considering independently each item when
maximizing its score.
Consider an inventory with n items must be classified based on J criteria.
yij : the measurement of the ith
item under the jth
criterion.
wij : the non- negative weight represents the contribution of performance of the ith
item for the jth
criterion.
Si : the score of the item i.
Phase I
Step 1. A pre-processing procedure:
Convert all measurement yij in 0 -1 scale.
Compute for each item i ( i = 1, . . . , n) a normalized measure using the following transformation:
 
   ijNiijNi
ijNiij
ij
yy
yy
y
,...1,...1
,...1*
minmax
min





Step 2. Weights generation procedure:
For each item i ( i = 1, . . . , n)
1,2......Jj0w
1
ij
1
*
1







J
j
ij
ij
J
j
iji
w
tosubject
ywSMax
J,.........,jx
j
z
tosubject
)x
j
(MaxzMinS
iji
ij,.....Jjii
21
1
1
1

 

4. An improvement of two multi-criteria inventory classification models
www.iosrjournals.org 24 | Page
Phase II
Step 1. Average weights computing
Jj
n
w
w
n
i
ij
ij .....,2,11


Step 2. Final scores
niywS
J
j
ijiji .....,2,1
1
*'
 
Step 3. The classification
Sort the scalar scores S’i in the descending order. Then Group the inventory items by the
principle of the ABC analysis
IV. Computational results
In order to illustrate and compare the performance of our proposed models to the R-Model and the
Ng-Model, we apply it to the MCIC problem using the 47 benchmark inventory items presented in Flores
et al (1992). Our experimental study is based on the following four criteria: average unit cost, annual dollar
usage, lead time and critical factor. More precisely, we add the critical factor criterion in the Ng-model
while considering the old dominance constraint and considering also that the critical factor is dominated by
the three others criteria. As in the previous studies we maintain the same distribution of 10-class A items, 14-
class B items and 23-class C items. It should be noted that all the considered criteria are positively related to
the importance degree called score of the inventory items.
IV.1. Performance of the R*model
In this section, we present a comparison between the R-model and the R*-model using
normalization. As shown in table 1, we note a difference between the two models. In fact, the percentage of
items correctly classified by the two models is 40%. The detailed benchmarking by class suggests differences
between the three classes in terms of percentage of items correctly classified by the two models. This
percentage is relatively high for class C (57%). For classes A and B respectively 30% and 21% of items are
similarly classified by the R-model and the R*-model.
The new classification according to the R*-model is structured as follows. The new class A is
composed by 3 items (15, 29 and 34) similarly classified by the old R-model. The remaining seven items
were over classified from class B to A (items 1 and 23) and from class C to A (items 9, 13, 32, 2 and 3).
The new class B in the R* model maintains the same 3 items 21, 31 and 40. Items 4, 14, 18, 19, 28 and 45
were under classified from class A to B. However, five items (10, 11, 12, 24 and 36) jumped from class C to B.
In the last class C, we find that 13 items are identically classified as in the R-model. Item 16 is dropped down
from class A to C. Nine other items (17, 22, 27, 33,37, 39, 47, 20 and 43) are under classified from class B to C.
The detailed obtained results are depicted in table 2.
IV.2. Performance of the M-Ng model
In this section, firstly we present a comparison between the Ng-model and the modified one. Then
we will analyze conflicts between R-model and the Ng-model that are solved by the M-Ng model. As shown
in table 3, we notice a significant difference between Ng and M-Ng models. In fact, the percentage of
items correctly classified by the two models is 60%. In the detailed benchmarking by class, we notice
relevant differences between the three classes in terms of percentage of items correctly classified by the two
models. This percentage is high for class C (74%) but decreases to 50% for items of class A and to 43% for
those of class B.
The five items: 18, 23, 21, 14 and 45 belonging to class B in the Ng-model jumped to the first class A when
using the modified Ng-model. The M-Ng model over classifies also six items (34, 12, 40, 22, 20 and 17) from
class C to the second class B. On the other hand, items 10 and 1 assigned to class A in the Ng model are
demoted to class B when using the new classifying model. Finally, we notice also in the modified Ng- model
that 6 items (5, 6, 4, 32,7 and 8) from classes A and B in the original Ng-model, are under classified in the
class C.
The obtained detailed results are depicted in table 4.

5. An improvement of two multi-criteria inventory classification models
www.iosrjournals.org 25 | Page
IV.3. Conflict handling
As shown in Table 4, there are 31 items that have been classified by the Ng-Model and the R-Model
in opposite classes. The proposed M-Ng model settles these conflicts by assigning a class suggested by one of
the two models in 29 cases out of 31 or giving a totally different class in 2 cases out of 31. In the first 29
handled conflicts, the M-Ng model chooses for 16 items 2, 3, 9, 13,15, 16, 19, 24, 27, 28, 33, 36, 37, 39, 43 and
47 the same classification given by the Ng-model, but chooses for the other 13 items 1, 5, 6, 7, 8, 14, 17, 18, 20,
22, 32, 40 and 45, the same class assignment given by the R-model.
On the other hand, for items 34 and 10 classified by the Ng-Model and the R-Model in op- posite classes A and
C, the M-Ng model attributes them the medium class B.
V. Conclusion
In this paper, improved models for multi-criteria inventory classification are proposed. In fact, the
first contribution of this work is to analyze the effect of using normalized data in the R-model. This new R*
model leads to 60% difference in items classification. The second contribution of this paper is the improvement
of the Ng model used in the MCIC by proposing a new model called M-Ng model. This new model
presents significant differences compared to initial Ng-model and also handles classification conflicts
between the R and Ng models. One way to extend this research is to apply and compare our improved models
using larger data sets for MCIC.
References
[1]. Chen J. 2011. Peer-estimation for multiple criteria ABC inventory classification, Computers & Operations Research, 38, 1784–
1791.
[2]. Flores, B. E., Olson, D. L., & Dorai, V. K. 1992. Management of multicriteria inventory classification. Mathematical and
Computer Modeling, 16,71–82.
[3]. Hadi-Vencheh A. 2010. An improvement to multiple criteria ABC inventory classification. European Journal of Operational
Research, 201, 962–965.
[4]. Ng W. L. 2007. A simple classifier for multiple criteria ABC analysis, European Journal of Operational Research, 177,
344–353.
[5]. Ramanathan R. 2006. ABC inventory classification with multiple-criteria using weighted linear optimization, Computers and
Operations Research,33, 695–700.
[6]. Zhou P., & Fan L. 2007. A note on multi-criteria ABC inventory classification using weighted linear optimization, European
Journal of Operational Research, 182, 1488–1491.
Table 1: Concordance between R and R* models classification
Class A Class B Class C Total
Number of items correctly classified
by the two models (R and R*) 3 3 13 19
Percentage of items correctly classified
by the two models (R and R*) 30% 21% 57% 40%
Table 2: A comparison of ABC classification using R and R* models
Initial data Normalized data ABC
classification
Item
no
Average
unit
cost($)
Annual
dollar
usage($)
Lead
time
Critical
factor
Average
unit
cost($)
Annual
dollar
usage
Lead
time
Critical
factor Score
R*
model
R
model
15
29
34
1
23
2
71,2
134,34
7,07
49,92
86,5
210
854,4
268,68
190,89
5840,64
432,5
5670
3
7
7
2
4
5
1
0,01
0,01
1
1
1
0,32
0,63
0,01
0,22
0,40
1,00
0,14
0,04
0,03
1,00
0,07
0,97
0,33
1,00
1,00
0,17
0,50
0,67
1,00
0,00
0,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
A
A
A
A
A
A
A
A
A
B
B
C
Continued on next page

6. An improvement of two multi-criteria inventory classification models
www.iosrjournals.org 26 | Page
Table 2 – continue
Initial data Normalized data ABC
classification
Item
no
Average
unit cost($)
Annual
dollar
usage($)
Lead
time
Critical
factor
Average
unit cost($)
Annual
dollar
usage
Lead
time
Critical
factor
Score
R*
model
R
model
3
9
13
32
4
14
18
19
28
45
21
31
40
10
11
12
24
36
16
17
20
22
27
33
37
39
43
47
5
6
23,76
73,44
86,5
53,02
27,73
110,4
49,5
47,5
78,4
34,4
24,4
72
51,68
160,5
5,12
20,87
33,2
40,82
45
14,66
58,45
65
84,03
49,48
30
59,6
29,89
8,46
57,98
31,24
5037,12
2423,52
1038
212,08
4769,56
883,2
594
570
313,6
34,4
463,6
216
103,36
2407,5
1075,2
1043,5
398,4
163,28
810
703,68
467,6
455
336,12
197,92
150
119,2
59,78
25,38
3478,8
2936,67
4
6
7
2
1
5
6
5
6
7
4
5
6
4
2
5
3
3
3
4
4
4
1
5
5
5
5
5
3
3
1
1
1
1
0,01
0,5
0,5
0,5
0,01
0,01
1
0,5
0,01
0,5
1
0,5
1
1
0,5
0,5
0,5
0,5
0,01
0,01
0,01
0,01
0,01
0,01
0,5
0,5
0,09
0,33
0,40
0,23
0,11
0,51
0,22
0,21
0,36
0,14
0,09
0,33
0,23
0,76
0,00
0,08
0,14
0,17
0,19
0,05
0,26
0,29
0,39
0,22
0,12
0,27
0,12
0,02
0,26
0,13
0,86
0,41
0,17
0,03
0,82
0,15
0,10
0,09
0,05
0,00
0,08
0,03
0,01
0,41
0,18
0,18
0,06
0,02
0,13
0,12
0,08
0,07
0,05
0,03
0,02
0,02
0,01
0,00
0,59
0,50
0,50
0,83
1,00
0,17
0,00
0,67
0,83
0,67
0,83
1,00
0,50
0,67
0,83
0,50
0,17
0,67
0,33
0,33
0,33
0,50
0,50
0,50
0,00
0,67
0,67
0,67
0,67
0,67
0,33
0,33
1,00
1,00
1,00
1,00
0,00
0,49
0,49
0,49
0,00
0,00
1,00
0,49
0,00
0,49
1,00
0,49
1,00
1,00
0,49
0,49
0,49
0,49
0,00
0,00
0,00
0,00
0,00
0,00
0,49
0,49
1,00
1,00
1,00
1,00
0,82
0,75
0,83
0,67
0,83
1,00
1,00
0,68
0,83
0,76
1,00
0,69
1,00
1,00
0,49
0,51
0,52
0,54
0,39
0,67
0,67
0,67
0,67
0,67
0,61
0,52
A
A
A
A
B
B
B
B
B
B
B
B
B
B
B
B
B
B
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
A
A
A
A
A
B
B
B
C
C
C
C
C
A
B
B
B
B
B
B
B
B
B
C
C
7
8
25
26
30
35
38
41
42
44
46
28,2
55
37,05
33,84
56
60,6
67,4
19,8
37,7
48,3
28,8
2820
2640
370,5
338,4
224
181,8
134,8
79,2
75,4
48,3
28,8
3
4
1
3
1
3
3
2
2
3
3
0,5
0,01
0,01
0,01
0,01
0,01
0,5
0,01
0,01
0,01
0,01
0,11
0,24
0,16
0,14
0,25
0,27
0,30
0,07
0,16
0,21
0,12
0,48
0,45
0,06
0,05
0,03
0,03
0,02
0,01
0,01
0,00
0,00
0,33
0,50
0,00
0,33
0,00
0,33
0,33
0,17
0,17
0,33
0,33
0,49
0,00
0,00
0,00
0,00
0,00
0,49
0,00
0,00
0,00
0,00
0,50
0,64
0,16
0,33
0,25
0,37
0,49
0,17
0,20
0,33
0,33
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Table 3: Concordance between Ng and M-Ng models classification
Class A Class B Class C Total
Number of items correctly classified
by the two models (Ng and M-Ng) 5 6 17 28
Percentage of items correctly classified
by the two models (Ng and M-Ng) 50% 43% 74% 60%

7. An improvement of two multi-criteria inventory classification models
www.iosrjournals.org 27 | Page
Table 4: A comparison of ABC classification using M-Ng, Ng and R models
ABC classification
Item
no
Average unit
cost($)
Annual
dollar
usage($)
Lead
time
Critical
factor
optimal
Inventory
score
M-Ng
Model
Ng
Model
R Model
2
3
9
13
14
18
21
23
29
0,97
0,86
0,41
0,17
0,15
0,10
0,08
0,07
0,04
1,00
0,09
0,33
0,40
0,51
0,22
0,09
0,40
0,63
0,67
0,50
0,83
1,00
0,67
0,83
0,50
0,50
1,00
1,00
1,00
1,00
1,00
0,49
0,49
1,00
1,00
0,00
0,817
0,631
0,802
0,881
0,564
0,622
0,573
0,602
0,606
A
A
A
A
A
A
A
A
A
A
A
A
A
B
B
B
B
A
C
C
C
C
A
A
B
B
A
45
1
10
12
15
17
19
20
22
24
28
31
34
36
40
4
5
6
7
8
11
16
25
26
27
30
32
33
35
37
38
0,00
1,00
0,41
0,18
0,14
0,12
0,09
0,08
0,07
0,06
0,05
0,03
0,03
0,02
0,01
0,82
0,59
0,50
0,48
0,45
0,18
0,13
0,06
0,05
0,05
0,03
0,03
0,03
0,03
0,02
0,02
0,14
0,22
0,76
0,08
0,32
0,05
0,21
0,26
0,29
0,14
0,36
0,33
0,01
0,17
0,23
0,11
0,26
0,13
0,11
0,24
0,00
0,19
0,16
0,14
0,39
0,25
0,23
0,22
0,27
0,12
0,30
1,00
0,17
0,50
0,67
0,33
0,50
0,67
0,50
0,50
0,33
0,83
0,67
1,00
0,33
0,83
0,00
0,33
0,33
0,33
0,50
0,17
0,33
0,00
0,33
0,00
0,00
0,17
0,67
0,33
0,67
0,33
0,00
1,00
0,49
0,49
1,00
0,49
0,49
0,49
0,49
1,00
0,00
0,49
0,00
1,00
0,00
0,00
0,49
0,49
0,49
0,00
1,00
0,49
0,00
0,00
0,00
0,00
1,00
0,00
0,00
0,00
0,49
0,556
0,473
0,517
0,524
0,510
0,427
0,531
0,444
0,447
0,486
0,490
0,538
0,546
0,487
0,475
0,071
0,392
0,373
0,370
0,328
0,391
0,352
0,019
0,198
0,041
0,026
0,402
0,385
0,209
0,375
0,354
A
B
B
B
B
B
B
B
B
B
B
B
B
B
B
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
B
A
A
C
B
C
B
C
C
B
B
B
C
B
C
A
A
A
B
B
C
C
C
C
C
C
B
C
C
C
C
A
B
C
C
A
B
A
B
B
C
A
B
A
C
B
A
C
C
C
C
C
A
C
C
B
C
C
B
C
B
C
39
41
42
43
44
46
47
0,02
0,01
0,01
0,01
0,00
0,00
0,00
0,27
0,07
0,16
0,12
0,21
0,12
0,02
0,67
0,17
0,17
0,67
0,33
0,33
0,67
0,00
0,00
0,00
0,00
0,00
0,00
0,00
0,388
0,098
0,106
0,374
0,201
0,192
0,363
C
C
C
C
C
C
C
C
C
C
C
C
C
C
B
C
C
B
C
C
B